Synchronization in oscillatory networks

Synchronization in oscillatory networks

M. Biey M. Bonnin P. Checco P.P. CivalleriF. Corinto M. Gilli V. Lanza M. Righero

Dipartimento di Elettronicahttp://lincs.delen.polito.it/

Politecnico di Torino, TORINO

XXIV RIUNIONE ANNUALEDEI RICERCATORI DI ELETTROTECNICA

Pavia, 19–21 giugno 2008

F. Corinto – ET 2008 – Pavia, 19th June 2008 Politecnico di Torino


Outline

(i) Weakly connected oscillatory networks (WCONs)

SynchronizationMathematical model

(ii) Global dynamic behavior

Joint application of Malkin’s theorem and of the describingfunction technique.Phase deviation equationCase study: Array of Chua’s circuits (Lur’e like model)

(iii) Applications

Detection of spatio-temporal patternsAssociative and Dynamic memories

(iv) Conclusions



Synchronization

Synchronization is the process of adjusting rhythms due to theweak interaction.

When subsystems (e.g. people, animals, cells, neurons)synchronize, they also can communicate.

Synchronous brain activity at about 40Hz is thought (Gammarhythms) to play an important role in the perception ofobjects (binding problem). Recent studies in neuroscience(Konig and Schillen) have shown that image elements couldbe coded by synchronized activity of cell assemblies inartificial oscillatory networks with local and delayed couplings.



The dynamics of coupled oscillators – Synchronization

Frequency lockingIzhik

evic

hand

Kura

moto

(2006)





evic

hand

Kura

moto

(2006)

Entrainmentωi = ω,∀i





evic

hand

Kura

moto

(2006)

Entrainmentωi = ω,∀i Phase locking





evic

hand

Kura

moto

(2006)

Entrainmentωi = ω,∀i Phase lockingSynchronization





evic

hand

Kura

moto

(2006)

Entrainmentωi = ω,∀i Phase lockingSynchronization

In-Phase

Anti-Phase

P. Checco, M. Biey, and L. Kocarev, ”Synchronization in random networks with given expected degree

sequences”, Chaos, Solitons and Fractals, 2006.

P. Checco, M. Biey, and M. Righero, ”Influence of Topology on Synchronization in Networks of Coupled

Hindmarsh-Rose Neurons”, ECCTD, Sevilla (Spain), August 26-30, 2007.



Oscillatory Networks

Single oscillator

Xi = Fi(Xi) Xi ∈ Rm, Fi : R

m → Rm, (i = 1, 2, ..., n)

has at least one hyperbolic Ti -periodic solution γi(t) : R → Rm

Γ

γ(t) S 1

θi(t) = ωi t, θi ∈ S1 = [0, 2π[ , ωi = 2 πTi

Weakly Connected Oscillatory Networks (ε ≪ 1)

Xi = Fi(Xi) + ε Gi(X ), X = [X ′

1, . . . X′n]

′, Gi : Rm×n → R

m

θi(t) = ωi t + φi (ǫt)



Global dynamic behavior – Malkin’s Theorem


Xi = Fi(Xi) + ε Gi(X ), X = [X ′1, . . . X

′n]

′, Gi : Rm×n → R

m


Time–domain techniques do not allow to identify all thelimit cycles (either stable or unstable).

It would require to consider infinitely many initial conditions.Unstable limit cycles cannot be detected through simulation.

By means of Spectral techniques (Describing Function andHarmonic Balance), the computation of all the limit cycles isreduced to a non-differential algebraic problem.

Such methods are not suitable for characterizing the globaldynamic behavior of complex networks with a large number ofattractors.



Global dynamic behavior – Malkin’s Theorem


Xi = Fi(Xi) + ε Gi(X ), X = [X ′1, . . . X

′n]

′, Gi : Rm×n → R

m


Phase deviation equation

φi =ω

T

∫ T

0Q ′

i (t) Gi

[

γ

(

t +φ − φi

ω

)]

dt,

T = m.c .m.(T1, . . . , Tn)

γ

(

t +φ − φi

ω

)

=

[

γ′

1

(

t +φ1 − φi

ω1

)

, . . . , γ′

n

(

t +φn − φi

ωn

)]′

Qi (t) = −[DFi(γi (t))]′Qi(t), Q ′

i (0)Fi (γi (0)) = 1



Joint application of the DF and MT

1 The periodic trajectories γi (t) of the uncoupled oscillators areapproximated through the describing function technique.

2 Once the approximation of γi (t) is known, a first harmonicapproximation of Qi (t) is computed, by exploiting the linearadjoint problem and the normalization condition.

3 The approximated phase deviation equation is derived byanalytically computing the integral expression given by theMalkin’s Theorem.



Joint application of the DF and MT

1 The periodic trajectories γi (t) of the uncoupled oscillators areapproximated through the describing function technique.

2 Once the approximation of γi (t) is known, a first harmonicapproximation of Qi (t) is computed, by exploiting the linearadjoint problem and the normalization condition.

3 The approximated phase deviation equation is derived byanalytically computing the integral expression given by theMalkin’s Theorem.

The phase equation is analyzed in order to determine the totalnumber of stationary solutions (equilibrium points) and theirstability properties. They correspond to the total number of limitcycles of the original weakly connected network.



Case study: 1-D array of n identical Chua’s circuits

xi = α [yi − xi − n(xi)] + ε (C−1 xi−1 + C+1 xi+1 − 2C0 xi )

yi = xi − yi + zi

zi = −β yi L C2 C1

R

b

b

b

b

iG(v)i

v2

+

−

v1

+

−

Xi =

xi

yi

zi

, Fi(Xi ) =

α [yi − xi − f (xi )]xi − yi + zi

−β yi

n(xi ) = −8

7xi +

4

63x3i , Gi (X ) =

C−1 xi−1 + C+1 xi+1 − 2C0 xi

00



Lur’e like WCONs

Li (D) xi(t) = n[xi(t)] + C−1 xi−1 + C+1 xi+1 − 2C0 xi

Li (D) =D3 + D2(1 + α) + Dβ + αβ

D2 + D + β

−4

−2

0

2

4

−3

−2

−1

0

1

2

3

−10

−5

0

5

10

xi(t)

yi(t)

z i(t)

Stable symmetric LC (Ssi)

Stable asymmetric LCs (A±i)

Unstable symmetric LC (Sui)

Uncoupled oscillator

α = 8, β = 15



Phase equation – DF approximation of limit cycles

Describing function approximation of γi (t)

xi(t) = Ai + Bi sin(ωt)

FAi

=1

π

∫ π

−π

−α n[Ai + Bi sin(θ)] dθ = −α Ai

(

−8

7+

4

63A2

i +2

21B2

i

)

FBi

=2

π

∫ π

−π

−α n[Ai + Bi sin(θ)] sin(θ) dθ = −α Bi

(

−8

7+

4

21A2

i +1

21B2

i

)

L(0) Ai = FAi (Ai ,Bi )

Re[L(jω)] Bi = FBi (Ai ,Bi )

Im [L(jω)] = 0

ω1 =

√

√

√

√

β −1 + α

2+

√

(

1 + α

2

)2

− β, ω2 =

√

√

√

√

β −1 + α

2−

√

(

1 + α

2

)2

− β



Phase equation – DF approximation of adjoint problem

Describing function approximation of Qi (t)

Qi (t) = −[DFi(γi (t))]′Qi(t), Qi(t) = [qi(t), . . . ]′

Q ′

i (0)Fi (γi(0)) = 1

qi (t) = δi cos(ωt), δi ∈ R

qi(t) is the only first harmonic expression of qi(t) satisfyingthe above equationδi is determined by imposing the normalization condition. Itdepends on the nature of the constituent oscillators (circuitparameters) and the periodic trajectory considered.



Phase equation – n identical oscillators with zero BC

WCONs with identical cells implies that Ai = A, Bi = B andωi = ω (entrainment),

φi =ω

T

∫ T

0Q ′

i (t) Gi

[

γ

(

t +φ − φi

ω

)]

dt

φi =ω

T

∫

T

0qi (t)

∑

k=±1

Ck xi+k

(

t +φi+k − φi

ω

)

dt =ω

T

∫

T

0δi cos(ωt)

∑

k=±1

Ck

[

A + B sin(

ωt + (φi+k − φi ))]

dt

φi = V (ω)∑

k=±1

Ck sin(φi+k − φi), V (ω) = ω δi B

Stationary solutions (synchronized states):(φj − φi) = {0, π} , i = 1, . . . , n, j = i ± 1



Phase equation – n identical oscillators with zero BC

WCONs with identical cells implies that Ai = A, Bi = B andωi = ω (entrainment),

φi =ω

T

∫ T

0Q ′

i (t) Gi

[

γ

(

t +φ − φi

ω

)]

dt

φi =ω

T

∫

T

0qi (t)

∑

k=±1

Ck xi+k

(

t +φi+k − φi

ω

)

dt =ω

T

∫

T

0δi cos(ωt)

∑

k=±1

Ck

[

A + B sin(

ωt + (φi+k − φi ))]

dt

φi = V (ω)∑

k=±1

Ck sin(φi+k − φi), V (ω) = ω δi B

Stationary solutions (synchronized states):(φi+1 − φi ) = {0, π} , i = 1, . . . , n − 1 ⇒ 2n−1 phase shifts



Total number of Limit cycles

For each one of the 2n−1 admissible phase shifts:

2n asymmetric limit cycles {A±

1 , A±

2 , ... A±n }. They correspond

to a frequency ω = ω2 and to a solution of the DF system with

sign(A1) = ±1, sign(A2) = ±1, ... sign(An) = ±1, respectively.

One symmetric limit cycle {S s1 , S s

2 , ... S sn−1, S s

n}. It corresponds

to a frequency ω = ω1 and to a solution of the DF system with

Ai = 0, ∀i .

One symmetric limit cycle {Su1 , Su

2 , ... Sun−1, Su

n }. It

corresponds to a frequency ω = ω2 and to a solution of the DF

system with Ai = 0, ∀i .

In accordance with the results obtained through the spectraltechniques (DF)



Total number of Limit cycles

Symmetriclimit cycles

withω = ω2:

2n−1


withω = ω1:

2n−1 Asymmetric limit cycles withω = ω2: 2n × 2n−1

V (ω2) < 0 and V (ω1) > 0



Limit cycle stability

J = V (ω)

−C+1 t1 C+1 t1 ... ... ... ...

C−1 t1 − [C−1 t1 + C+1 t2] C+1 t2 ... ... ...

... ... ... ... ... ...

... ... ... ... ... ...

... ... ... C−1 tn−2 − [C−1 tn−2 + C+1 tn−1] C+1 tn−1

... ... ... ... C−1 tn−1 −C−1 tn−1

ti = cos(φi+1 − φi )

The whole dynamical system exhibits n × m Floquet’smultipliers µ [Characteristic exponents, λ : µ = exp(λT )].

The eigenvalues of the Jacobian matrix yields an accurateestimation of the Coupling Characteristic Exponents (i.e. then exponents that in absence of coupling equal zero.)




Proposition

Let us assume C−1/C+1 > 0, which implies that all the eigenvaluesof the matrix J are real. Let us denote with N the number ofeigenvalues of J that have the same sign of V (ω)C+1, with M thenumber of eigenvalues of J that have opposite sign to V (ω)C+1,and with L the number of null eigenvalues. Then L = 1, N equalsthe number of phase shifts ηi = π, and M equals the number ofphase shifts ηi = 0.

If V (ω)C+1 > 0 (V (ω)C+1 < 0), then the number of stableCoupling FMs equals the number of phase shifts ηi = 0(ηi = π); the number of unstable Coupling FMs equals thenumber of phase shifts ηi = π (ηi = 0).




Conjecture

Let us assume C−1/C+1 < 0. Let us denote with N the number ofeigenvalues of J, whose real part has the same sign ofV (ω) (C−1 + C+1), with M the number of eigenvalues of J whosereal part has opposite sign to V (ω) (C+1 + C−1), and with L thenumber of null eigenvalues. Then L = 1, N equals the number ofphase shifts ηi = π, and M equals the number of phase shiftsηi = 0.

If V (ω) (C+1 + C−1) > 0 (V (ω) (C+1 + C−1) < 0), then thenumber of stable Coupling FMs equals the number of phaseshifts ηi = 0 (ηi = π); the number of unstable Coupling FMs

equals the number of phase shifts ηi = π (ηi = 0).




Asymmetric limit cycles: (V (ω2 ) < 0 )

There are 2n stable asymmetric limit cycles, i.e. thosecorresponding to {A±

1 , A±

2 , ... A±

n−1, A±n }, with all the phase

shifts equal to π (ηi = π, 1 ≤ i ≤ n − 1).

The other 2n × (2n−1 − 1) asymmetric limit cycles areunstable; they present as many Floquet’s multipliers |µ| > 1as the number of phase shifts ηi = 0.




Symmetric limit cycles

There is only one stable symmetric limit cycle withV (ω1 ) > 0 . It corresponds to {S s

1 , S s2 , ... S s

n−1, S sn}, with all

the phase shifts equal to zero (ηi = 0, 1 ≤ i ≤ n − 1).

There are 2n−1 − 1 unstable symmetric limit cycles withV (ω1 ) > 0 . They correspond to {S s

1 , S s2 , ... S s

n−1, S sn}, with

at least one phase shift equal to π (∃ i: ηi = π). They exhibitas many Floquet’s multipliers |µ| > 1 as the number of phaseshifts ηi = π.

There are 2n−1 unstable symmetric limit cycles withV (ω2 ) < 0 , corresponding to {Su

1 , Su2 , ... Su

n−1, Sun }. The

number of Floquet’s multipliers with |µ| > 1 can be computedas n plus the number of phase shifts ηi = 0.



Total number of Limit cycles and their stability


withω = ω2:

2n−1


withω = ω1:2n−1 − 1

One stable symmetric limit cycle

Asymmetric limit cycles withω = ω2: 2n × (2n−1 − 1)

Asymmetric limit cycles withω = ω2: 2n

V (ω2) < 0 and V (ω1) > 0



Seven cells: Asymmetric limit cycle

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

Appppppp000000 [0.01, 0.1, C+1]

Appppppp000000 [−0.01, 0.1, C+1]

Appppppp000ππ0 [0.01, 0.1, C+1]

Appppppp000ππ0 [−0.01, 0.1, C+1]



Seven cells: Symmetric limit cycle

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

1

0.01 |

|µ|

C+1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

0.01

1

|µ|

C+1

Ss000000 [0.01, 0.1, C+1]

Ss000000 [−0.01, 0.1, C+1]

Ss000ππ0 [0.01, 0.1, C+1]

Ss000ππ0 [−0.01, 0.1, C+1]



Applications

Synchronous states can be exploited for dynamic patternrecognition and to realize associative and dynamic memories.By means of a simple learning algorithm, the phase-deviationequation is designed in such a way that given sets of patternscan be stored and recalled. In particular, two models ofWCONs have been proposed as examples of associative anddynamic memories. (ET2007)

Spiral waves are the most universal form of patterns arising indissipative media of oscillatory and excitable nature. Byfocusing on oscillatory networks, whose cells admit of a Lur’edescription and are linearly connected through weak couplings,the occurrence of spiral waves has been studied. (ISCAS 2008)



References

M. Bonnin; F. Corinto; M. Gilli, ”Periodic Oscillation In Weakly ConnectedCellular Nonlinear Networks”, IEEE Trans. Circuits Syst. I, July 2008, ISSN:1057-7122, DOI: 10.1109/TCSI.2008.916460

V. Lanza; F. Corinto; M. Gilli; P. P. Civalleri, ”Analysis of nonlinear oscillatorynetwork dynamics via time-varying amplitude and phase variables”, INT J CIRC

THEOR APP, pp. 623-644, 2007, Vol. 35, ISSN: 0098-9886, DOI:10.1002/cta.v35:5/6

F. Corinto; M. Bonnin; M. Gilli, ”Weakly Connected Oscillatory Network Modelsfor Associative and Dynamic Memories”, INT J BIFURCAT CHAOS, pp.4365-4379, 2007, Vol. 17, ISSN: 0218-1274

M. Bonnin; F. Corinto; M. Gilli, Bifurcations, ”Stability And Synchronization inDelayed Oscillatory Networks”, INT J BIFURCAT CHAOS, pp. 4033-4048,2007, Vol. 17, ISSN: 0218-1274

M. Gilli, F. Corinto, P. Checco, ”Periodic oscillations and bifurcations in cellularnonlinear networks”, IEEE Trans. Circuits Syst. I, pp. 948-962, 2004, Vol. 51,ISSN: 1057-7122, DOI: 10.1109/TCSI.2004.827627

web link: http://lincs.delen.polito.it/



Conclusions

Weakly connected oscillatory networks

Periodic oscillations and global dynamic behavior.

* Analytical results through the joint application of the Malkin’sTheorem and the describing function technique.

* Phase deviation equation for generic one/two dimensional,locally/fully connected, space invariant/variant networks ofoscillators.

* Equilibrium points → Limit cycles.* Equilibrium point stability → Limit cycle stability.* Total number of limit cycles, with their stability characteristics.

Detection of spatio-temporal patterns

Associative and Dynamic memories



Synchronization in oscillatory networks

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